A204270 a(n) = tau(n)*Pell(n), where tau(n) = A000005(n), the number of divisors of n.

1, 4, 10, 36, 58, 280, 338, 1632, 2955, 9512, 11482, 83160, 66922, 323128, 780100, 2354160, 2273378, 16465260, 13250218, 95966568, 154455860, 372889432, 450117362, 4346717760, 3935214363, 12667263848, 30581480180, 110745336312, 89120964298

Offset

1,2

Comments

Compare g.f. to the Lambert series identity: Sum_{n>=1} x^n/(1-x^n) = Sum_{n>=1} tau(n)*x^n.

Related identities:

(1) Sum_{n>=1} n^k*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_{k}(n)*Pell(n)*x^n for k>=0.

(2) Sum_{n>=1} phi(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Pell(n)*x^n.

(3) Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = x.

(4) Sum_{n>=1} lambda(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Pell(n^2)*x^(n^2).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{n>=1} Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} tau(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

Example

G.f.: A(x) = 1 + 4*x + 10*x^2 + 36*x^3 + 58*x^4 + 280*x^5 + 338*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2*x^2/(1-6*x^2+x^4) + 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) + 29*x^5/(1-82*x^5-x^10) + 70*x^6/(1-198*x^6+x^12) +...+ Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Mathematica

Table[DivisorSigma[0, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)

Prog

(PARI) /* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}
(PARI) {a(n)=sigma(n, 0)*Pell(n)}
(PARI) {a(n)=polcoeff(sum(m=1, n, Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

Crossrefs

Cf. A203847, A204271, A204272, A204273, A204274, A204275, A000005 (tau), A002203, A000045.

Sequence in context: A152916 A222506 A108596 * A287220 A220205 A196880

Adjacent sequences: A204267 A204268 A204269 * A204271 A204272 A204273

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2012

Status

approved